The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families

Let $\omega (\mathcal{F})={\sum }_{\{A,B\}\subset \mathcal{F}}\mathrm{\mid }A\cap B\mathrm{\mid }$ and $\omega (\mathcal{A},\mathcal{B})={\sum }_{(A,B)\in \mathcal{A}\times \mathcal{B}}\mathrm{\mid }A\cap B\mathrm{\mid }$. A family $\mathcal{F}$ is intersecting if $F_1\cap F_2\mathrm{\ne }\mathrm{\varnothing }$ for any $F_1,F_2\in \mathcal{F}$ and two family $\mathcal{A}$ and $\mathcal{B}$ are crossing-intersecting if $A\cap B\mathrm{\ne }\mathrm{\varnothing }$ for any $(A,B)\in \mathcal{A}\times \mathcal{B}$. For an intersecting family $\mathcal{F}$, Erdős, Ko and Rado determined the upper bound of $\mathrm{\mid }\mathcal{F}\mathrm{\mid }$, consequently yielding an upper bound of $\left(\genfrac{}{}{0px}{}{\mathrm{\mid }\mathcal{F}\mathrm{\mid }}{2}\right)={\sum }_{\{A,B\}\subset \mathcal{F}}1$. If we replace $1$ with $\mathrm{\mid }A\cap B\mathrm{\mid }$ in the summation ${\sum }_{\{A,B\}\subset \mathcal{F}}1$, then this summation transforms into $\omega (\mathcal{F})$. In this paper, for an intersecting family $\mathcal{F}$, we determine the upper bound of $\omega (\mathcal{F})$, which is a generalization of Erdős-Ko-Rado Theorem. Further, for crossing-intersecting families $\mathcal{A}$ and $\mathcal{B}$, we determine the upper bound of $\omega (\mathcal{A},\mathcal{B})$.